On identities in matrix groups (Q1896649)

It is proved that in the groups \(\text{Gl}(n, K)\), \(\text{Sl} (n,K)\) \((K\) is a finite or infinite field) no nontrivial identity \(A_1 X^{k_1} A_2 X^{k_2} \dots A_m X^{k_m} =1\) holds for all \(X\) in the group, \(A_1, \dots, A_m\) not in the center of the groups -- under the assumption that the cardinality of \(K\) is greater than \(n\) and \(2(k_1 + \cdots + k_m)\).